AN ABSTRACT OF THE THESIS OF
in
Daniel K. Melchior
for the degree of Master of Science
Mechanical Engineering
presented on
Title:
March 12, 1980
The Effect of Geometric Orientation and Random Wind
Conditions on Flat Plate Convection Rates
Redacted for Privacy
Abstract approved:
Professor Milton B. Larson
An experimental investigation has been performed to determine the
forced convection heat transfer due to three-dimensional air flow over
a flat plate as a function of angle of attack and free stream velocity.
The experimental method employed a transient thermal technique and
placed the convecting surface next to the exhaust port of an open
circuit wind tunnel.
The ratio of convecting surface width to wind
tunnel nozzle width was 0.7, but insulating surfaces extended beyond
the nozzle width.
The flows evaluated ranged in Reynolds number from
32,000 to 140,000 and from zero to 90-degree angles of attack.
The
data indicated the probable existence of a turbulent boundary layer
at zero and 45-degree angles of attack.
Data collected for angles of
attack of 30, 60, and 90 degrees demonstrate mostly laminar boundary
layer flows and indicate a dependence of the Colburn i-factor upon
angle of attack.
The j-factor was determined to be proportional to
the Reynolds number to the minus one-half power, as in previous work,
and the multiplier varied between 0.876, as in a Pohlhausen equation
corrected for unheated starting length, to 1.114 as predicted in
wedge flow measurements performed by Eckert.
The results are differ-
ent from those of Sparrow and Tien, which indicate a single multiplier
of 0.931 will provide solutions for all angles of attack between 25
and 90 degrees.
The flow in this investigation is largely two-
dimensional due to the physical arrangement of the apparatus, whereas
the flow used by Sparrow and Tien was primarily three-dimensional.
An analytical method was also developed which accounted for the
random nature of the natural wind velocity distribution in the calculation of the time average convection film coefficient.
The method
uses the average velocity and the standard deviation of the velocity
to provide an evaluation of the time average convection film coefficient.
This was accomplished by transforming the defining convection
relationship into a second order constant coefficient polynomial over
a fixed range of velocities.
The method was demonstrated on a statis-
tical record of wind velocities measured on November 29 and 30, 1978 at
the Oregon State University campus.
The error created by using only
the average velocity in calculating the time average convection film
coefficient was determined to be 3.4%.
The evaluation concludes
that with present uncertainties in the convection film coefficient,
the error introduced by ignoring the random nature of the wind is
insignificant.
0 Copyright by Daniel K. Melchior
February 29, 1980
All Rights Reserved
THE EFFECT OF GEOMETRIC ORIENTATION
AND RANDOM WIND CONDITIONS ON FLAT
PLATE CONVECTION RATES
by
Daniel K. Melchior
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Commencement June 1980
APPROVED:
Redacted for Privacy
Professor of Mechanical Engineering in charge of major
Redacted for Privacy
Head of
ical Engineering Depar
ent
Redacted for Privacy
\Dean of Gr duate School
Date thesis is presented
1
March 12, 1980
Typed by Donna Lee Norvell for
Daniel K. Melchior
ACKNOWLEDGEMENTS
The author wishes to acknowledge the endless support,
understanding, and encouragement of his wife, Pamela,
and daughter, Kasee.
TABLE OF CONTENTS
I.
II.
INTRODUCTION
DETERMINATION OF THE CONVECTION FILM COEFFICIENT
BY EXPERIMENTAL METHODS
2.1
2.1.1
2.1.2
2.1.3
2.2
6
Experimental Methods Used
Controlled heat transfer
Transient heat transfer
Mass transfer analogies
steady state
6
7
11
13
Experimental Apparatus
2.2.1
13
Individual component design
Fan
Plate
Insulation block
2.2.2
2.3
2.3.2
2.3.3
2.4.3
Heating and stabilizing plate temperature.
Measurement of temperature and time
Measurement of air velocity
.
Mathematical model and analysis
Experimental data
Uncertainty analysis
Results and Comparison with Previous Work
2.5.1
III.
23
Data Reduction
2.4.1
2.4.2
2.5
20
Experimental Procedure
2.3.1
2.4
Means of orientation
System integration
Applicability to solar collectors
23
25
26
29
29
33
33
35
45
EXTENDING THE CONVECTION FILM COEFFICIENT DETERMINED FOR
CONSTANT VELOCITIES TO RANDOM VELOCITY DISTRIBUTIONS,
AN APPROXIMATE ANALYTICAL APPROACH
3.1
Evaluating the Covection Film Coefficient Using_
the Average and the Standard Deviation of the
Wind Velocity
49
3.2
IV.
Examination of the Error Introduced by Using the
Average Wind Velocity to Predict the Average
Convection Film Coefficient
55
CONCLUSIONS
4.1
4.2
4.3
Dependence of the Convection Film Coefficient
Upon Angle of Attack
61
Random Wind Velocities and the Average Convection
Film Coefficient
63
Future Research
65
BIBLIOGRAPHY
APPENDICES
Appendix A -- Nomenclature
Appendix B -- Determination of Stable Temperature
Decay Region (Sphere Tests)
Appendix C -- Determination of Insulation Losses.
Appendix D -- Tabulated Data and Results
Appendix E -- Tables, Charts, and Graphs
68
70
.
74
79
80
LIST OF ILLUSTRATIONS
Page
Figure
1
Measured thermal conductance of insulation block.
assembly
10
.
2
Wind tunnel design
15
3
Wind tunnel sketch
16
4
Exit port nozzle design
17
5
Insulation block assembly (cut-away view)
19
6
System arrangement
21
7
Test configuration.
22
8
Method used to warm aluminum plate
24
9
Typical time/temperature profile
27
10
System energy balance
30
11
Results of previous zero degree angle of
attack studies
36
12
Zero degree angle of attack data
38
13
30-degree angle of attack data
40
14
45-degree angle of attack data
41
15
60-degree angle of attack data
43
16
90-degree angle of attack data
44
17
Measured j-factor dependence upon angle of attack
18
Agreement between interpolating polynomial and
governing equation (3-23)
19
20
Measured temperature dependence of the flat plate
convection film coefficient
Measured temperature dependence of the sphere
convection film coefficient
.
.
46
57
71
72
List of Illustrations, continued
Page
Figure
Total insulation block assembly thermal conductance
as a function of plate temperature
77
22
Fan performance
83
23
Wind velocity measurement uncertainty
86
Design constraints for wind tunnel
14
Values of parameters used for data reduction with
equation (2-15)
34
3
Values used for parameters in equation (3-22)
55
4
Selected points
56
5
Errors introduced by ignoring statistical information.
58
6
Measured decay times and calculated conductance losses
75
7
Total measured conduction losses through insulation
block assembly
76
8
Measured data
79
9
Calculated j-factors
80
10
Biot numbers for various materials
82
11
Properties of the aluminum plate
84
12
Properties of insulation block material
85
13
Wind velocity data
87
21
Table
1
2
THE EFFECT OF GEOMETRIC ORIENTATION
AND RANDOM WIND CONDITIONS ON FLAT
PLATE CONVECTION RATES
SECTION I
INTRODUCTION
2
INTRODUCTION
The challenge of meeting our nation's growing energy demands requires the prudent use of our energy resources.
Our increasing re-
quirements coupled with ever-decreasing supplies of useful energy
and the associated economic inflation dictates tighter engineering
tolerances for our energy using and producing systems.
These tighter
tolerances are required as our need for more optimal designs increases.
Large safety factors built into the space heating systems in most
buildings, which account for unpredictable or unknown energy losses,
reduce the overall system efficiency.
Solar collector arrays are over-
sized to account for unknown top losses due to ambient wind effects.
These are just two examples of many engineering systems which are oversized because of imprecise knowledge of ambient effects on surface
heat transfer.
Forced convection heat transfer has been the subject of much research throughout modern history.
However, little of the research is
directly applicable to modern flat plate solar collectors.
Early work performed by Dr. Ing-Walter Jurges [1] of Germany in
1924 has provided the basis for the wind-dependent convection film
coefficient predicting equation used in most of the current solar
collector top loss calculations.
The validity of using Jurges' equa2
tion (1-1), where his expressed in w/m °C and V is in m/s,
h
w
= 5.7 + 3.8V
3
for solar collector top loss calculations is questionable.
The in-
vestigation evaluated only a moving air stream parallel to a flat
plate, and ignores the dependence of the convection film coefficient
upon the distance from the leading edge.
These questions prompted
E. M. Sparrow and K. K. Tien [2][3] to undertake an investigation
which evaluated the effect of various angles of attack upon the convection film coefficient.
Their investigation used a mass sublimation
technique to infer the heat transfer rate.
The apparatus consisted of
a 7.6 cm square plate suspended at various angles of attack within a
wind tunnel.
The maximum ratio of wind tunnel width to plate width
was 0.25 and blockage was determined to be negligible.
The air flow
was relatively unrestricted and angles of attack between 25 and 90 degrees were investigated for Reynolds numbers between 20,000 and
100,000.
The results of their work, shown in equation (1-2), demon-
strated virtual independence of the convection film coefficient, as
reflected in the Colburn j-factor, with respect to angle of attack.
(1-2)
j = .931 Re-1/2
Recent work by R. G. Sam, R. C. Lessmann, and F. L. Test [4]
from the University of Rhode Island on the pressure distribution on
the surface of a rectangular prism for various angles of attack and
wind velocity currents investigated laminar and turbulent twodimensional flow characteristics.
The report provided information
as a part of an investigation into convective film coefficients for
4
inclined surfaces as a function of wind currents.
The heat transfer
work, sponsored by the National Science Foundation, is currently in
progress.
The present investigation provides information which bridges
the earlier work of Jurges and the recent work of Sparrow and Tien.
The method used consisted of a transient thermal technique in which
a flat plate of known specific heat was warmed and allowed to cool
under controlled ambient conditions.
The physical apparatus used a
30.4 cm square plate mounted in an insulation block and oriented at
various angles of attack relative to the exhaust port of an open
circuit wind tunnel.
width was 0.7,
The ratio of convecting surface width to nozzle
while the ratio of the insulated surface width to
the nozzle width was 1.35.
The wind velocities ranged between 1.5
and 7.6 m/s (5 to 25 ft/sec) which correspond to a Reynolds number
range from 32,000 to 140,000.
and 90 degrees.
Angles of attack ranged from zero
5
Section II
DETERMINATION OF THE CONVECTION FILM
COEFFICIENT BY EXPERIMENTAL METHODS
6
2.1
EXPERIMENTAL METHODS USED
The convection film coefficient can be experimentally measured
in many different ways.
transport phenomenon.
Most of the techniques rely on two different
The first group relies on direct measurement
of heat transfer and may use steady state systems or transient techniques.
The second group relies on the measurement of mass transfer
as an indirect measurement of heat transfer using the analogy between
mass and heat transfer to infer the convection film coefficient.
Various advantages and disadvantages of both measurement techniques
will be discussed in the following section along with experimental
methods which were found to reduce the error associated with the direct transient heat transfer technique employed in this investigation.
2.1.1
Controlled Heat TransferSteady State
A steady state controlled heat transfer method was used by W.
Jurges [1] in 1924.
This technique involves the establishment of a
uniform heat flux interface at the heat transfer surface.
The energy
applied to the plate in the form of electric resistance heating was
controlled to maintain a 30°C temperature difference between the plate
surface temperature and the ambient while exposed to a steady and established wind velocity relative to the plate.
Losses through the in-
sulation block were measured and deducted from the total energy supplied to the system to determine the rate at which energy was convected from the surface.
This method is obviously the most direct and
7
with proper experimental controls should provide very accurate results.
It should be noted, however, that the experiment performed by
Jurges only examined conditions of zero degree angle of attack.
This
method would be ideal for measuring the effect of angle of attack, and
with the use of modern strip heaters and insulation materials should
provide excellent results.
A major disadvantage of this technique is
the length of time required to establish a steady state convection
situation.
The stabilizing time can be decreased by designing for a
small thermal mass and automatic temperature control.
2.1.2
Transient Heat Transfer
The transient heat transfer method, such as that used in this
investigation, consists of heating a material of a known specific
heat to a high temperature and then allowing the material to cool to
ambient temperature under controlled conditions.
Measurement of the
rate of temperature decay will yield the convection film coefficient
directly in terms of heat transfer for the specific conditions experienced during the temperature decay.
While this method is the most
direct and the easiest to implement, it does have characteristics
which limit the accuracy which can be achieved.
Meaningful results
can only be achieved if the entire body undergoing a thermal decay is
at a uniform or nearly uniform temperature.
The mathematical complexi-
ties introduced by allowing for a nonuniform temperature distribution
throughout the body are impractical to implement.
However, a nearly
uniform temperature throughout the body can be achieved if the
8
conduction resistance in the solid is much less than the convection
resistance at the surface of the body.
This ratio has been given the
name of the Biot number (Bi) and is a quantitative representation of
the degree of temperature uniformity achievable.
General practice has
indicated that an error of less than 5% in uniformity of temperature
is achieved for Biot number less than 0.1.
The mathematical expression
which describes the Biot number is shown in equation (2-1) where the
convection film coefficient and the thermal conductivity of the material are represented by h and k, respectively.
Bi = kQ
(2 -1 )
The characteristic length Q is often found by dividing the volume of
the body by the convecting surface area, which for a flat plate with
Care must be taken to
one exposed surface will yield the thickness.
express h, k, and R. in the same system of units.
The Biot numbers
for several materials are tabulated in Appendix E for the maximum ex2
pected convection film coefficient of 19 w/m °C at 7.6 m/s wind velocity and a plate thickness of 0.406 cm.
search was 7075 T6 Alclad
The material used in this re-
aluminum plate.
The present results indicate that only a specific range of temperature difference between plate and ambient temperature may be used
to evaluate the convection film coefficient.
Large temperature dif-
ferences between the plate and ambient result in greater thermal losses through the supporting insulating materials.
When the plate
9
temperature approaches ambient, very small temperature differences between where the thermocouple is located and the surface can cause
large deviations in the derived convection film coefficient.
Direct heat transfer measurement techniques suffer the common
problem of unknown or uncontrollable thermal losses through means other
than the forced convection under investigation.
These losses can be
due to conduction through the insulation material or changes in the
internal energy of the insulation.
Conduction losses through the in-
sulated surfaces may be dependent upon the temperature of the plate
due to possible temperature-dependent characteristics of the insulation.
A method was developed which allowed an experimental approximation of
the conduction losses as a function of the plate temperature using the
same transient thermal technique which was used to determine the convection film coefficient.
The procedure is explained in detail in
Appendix C and the results are shown in Figure 1 and Appendix C.
The experimenter must be careful to use only data obtained when
the assumption of one homogenous temperature is sufficiently accurate.
The experimental determination of the temperature regions in which the
assumption is sufficiently accurate is explained in detail in
Appendix B.
0.13
0.08
(0.15)
Apparatus
Conductance
WPC
(BTU/hr°F)
0.05
(0.10)
(110)43
(130 54
(150)66
(170)78
Temperature(°F) °C
190 88
MEASURED THERMAL CONDUCTANCE OF INSULATION BLOCK ASSEMBLY
FIGURE 1
210)99
11
2.1.3
Mass Transfer Analogies
The mass transfer technique employed by Sparrow and Tien [2][3]
involved the measurement of the sublimation of naphthalene from a
7.62 cm (3 in) square plate oriented at various angles of attack and yaw
within a low-velocity, low-turbulence wind tunnel.
According to
Sparrow, the average mass transfer is related to the average heat
transfer in such a way that their associated j-factors are equal. The
j-factor for mass transfer and heat transfer are shown in equation
(2-2) and equation (2-3), respectively.
ent, k', is defined in equation (2-4).
The mass transfer coefficiAppendix A defines the nomen-
clature used throughout this paper.
(k7u.)Sc2/3
j
=
j
= StPr
2/3
(mass transfer)
(2-2)
(heat transfer)
(2-3)
(2-4)
k' = M(pnw - pno,)
Mass transfer methods can yield greater precision than direct
heat transfer measurement techniques because surfaces which are ideally abiabatic in the heat transfer problem may be accurately represented
by surfaces composed of a nonsubliming material.
The integrated aver-
age mass transfer rate is directly obtained through the calculation
of the mass sublimed in a given time period.
Such a calculation only
required the difference between the starting and ending mass and
the elapsed time.
The greatest uncertainty results from the mass
12
sublimed during the time required to set up and dismount the sample.
It is important to note that the accuracy of any mass transfer
method is dependent upon the accuracy of the mass to heat transfer
analogy.
13
2.2
EXPERIMENTAL APPARATUS
The apparatus used in the experimental determination of the convection film coefficient consisted of a low-velocity wind tunnel, a
square aluminum plate mounted in an insulation block with a means for
orientation to predetermined angles relative to the wind tunnel air
stream, and an Esterline Angus PD 2064 Data Logger.
The various com-
ponents will be described in detail and a description of the integrated system layout will follow.
2.2.1
Individual Component Design
The desired properties of the wind source are described in Table
3.
The design conceived to meet the criteria listed in Table 1 is
shown in Figure 2 and Figure 3.
The wind tunnel consisted of two
50.8 cm (20 in) axial fans along a common axis.
Each fan had three
individual speed settings as well as an easily damped intake port allowing for a wide range of exit velocities.
The performance of the
fan as a function of exit area is shown in Appendix E.
of 1096 cm
2
An exit area
(170 in2) was found to offer the best performance and also
provide a uniform velocity air stream over the test plate.
The ex-
haust port nozzle was designed to provide a gradual transition from
the circular pipe to the square exit area.
Care was taken to make the
convergence gradual and to provide a small straightening section in
the exhaust grill to correct for any distortion due to the circular
to square transition.
The exhaust port was designed such that the
square exit area could be rotated to any desired angle if future tests
14
would require a different configuration.
The design for the exhaust
port nozzle is shown in Figure 4.
TABLE 1
DESIGN CONSTRAINTS FOR WIND TUNNEL
1.
Velocity variable from 1.5 to 7.6 m/s (5 to 25 ft/sec)
2.
Provide a constant and predictable velocity at the exhaust
port which was uniform throughout the cross-section (square
cross-section).
3.
Have a large enough exit area such that the wind velocity
would be uniform over the surface of the plate.
4.
Provide a wind source which could be.external to the heat
transfer surface and allow unrestricted three-dimensional
flow above the heat transfer surface.
Testing of the velocity distribution over the cross-section of
the exit port indicated considerable spiral of the air stream in the
pipe section between the fans and the exit port.
This condition was
corrected by installing 30.50 cm x 10 cm diameter (12 in x 4 in diameter) tubes just behind the diffuser section in the large pipe area as'
indicated in Figure 2.
Velocity distribution tests after installation
of the straightening tubes showed a maximum variation in exit port
velocity of 2% over the cross-section.
This variation was thought to
be the best which could be achieved with the funds available and was
consistent with accuracy expected on a system level.
intensity in the exit air stream was not determined.
The turbulence
1.98
(78.00)
Straightening
Section
Flow Stabilization
Section
OOOOOOO
Dimensions in (in) m
WIND TUNNEL DESIGN
FIGURE 2
Exit Nozzle
Port
WIND TUNNEL SKETCH
FIGURE 3
0.5(19.75)Dia.
0.25
(10.00)
(17000)
0.43
(7.00)
____
0.178
m
Dimensions in (in)
EXHAUST PORT NOZZLE DESIGN
FIGURE 4
or-43.00)
0.076
18
The source of thermal energy in the experimental apparatus was a
30.48 x 30.48 x .406 cm (12 x 12 x .160 in) aluminum plate.
perties of the plate material are specified in Appendix E.
The proThe mass
of the plate was determined by weighing the plate to within one-half
gram.
Holes for thermocouples were drilled into the rear surface of
the plate to within 0.1 cm (.040 in) of the front surface.
This ar-
rangement left a 13.97 cm (5.5 in) border of insulation around the
square aluminum plate and allowed the surface of the aluminum plate to
be flush with the surface of the aluminum foil-faced insulation material.
A mounting plate which would accept the threaded stud of a camera
tripod was attached to the
adhesive.
back
of the insulation block with contact
The resulting assembly allowed the convenient adjustment of
the insulation block and the associated heat transfer surface to any
desired angle.
Prior to the final installation of the aluminum plate
into the insulation block, a small hole was drilled through the insulation and mounting plate to allow the thermocouple wires to be fed
Once the aluminum plate was
through the back of the insulation block.
installed in the insulation block, the interface between the edge of
the aluminum plate and the insulation block was taped with hightemperature tape to assure a smooth aerodynamic transition between
The final assembly allowed
the plate and the foil-faced insulation.
orientation of the exposed surface of the plate to any desired angle.
The construction of the aluminum plate and the installation into the
insulation block are shown in Figure 5.
The specified insulative
properties of the insulation material and the measured insulation
block conductance are shown in Appendix E.
Convecting Surface
(aluminum plate)
Thermocouple
locations
Insulation
Aluminum
Mounting plate
Camera tripod
mounting hole
Thermocouple wire
access hole
INSULATION BLOCK ASSEMBLY (CUTAWAY VIEW)
FIGURE 5
tO
20
2.2.2
System Integration
The integrated system is shown in Figure 6.
The purpose of the
low-velocity wind tunnel is to provide a steady horizontal air flow
over the convecting surface.
The surface can be adjusted to any desired
angle of impingement through the operation of the camera tripod mount.
Once the air flow leaves the exhaust port of the wind tunnel, the
stream is free to interact with the heat transfer surface in a threedimensional manner and is restricted only by the interaction with the
insulation block.
Figure 7 shows the test configurations used.
This
arrangement should provide realistic data which are applicable to many
real-life engineering situations such as solar collectors, building
walls, and roofs.
Convecting
Surface
a.)
N
N
Fan
Wind Tunnel
0
#2
Insulation Block
Camera
Tripod
Table
SYSTEM ARRANGEMENT
FIGURE 6
Fan
#1
Nozzle
Straightening
Tubes
Air
Angle
of Attack
Flow
TEST CONFIGURATIONS
FIGURE
7
23
2.3
EXPERIMENTAL PROCEDURE
The experimental procedure used in a transient heat transfer
method, such as the one employed in this investigation, must be carefully controlled to assure maximum accuracy.
The following section
covers the experimental techniques developed throughout the
course of this investigation which enhanced the accuracy and repeatability of the data collected.
Three main areas which require special
attention are heating and stabilizing the aluminum plate temperature,
measurement of the decaying temperature and associated times, and the
accurate measurement of the air velocity.
2.3.1
Heating and Stabilizing Plate Temperature
The mathematical model used to analyze the data requires the
assumption of a uniform plate temperature.
Any deviation from a uni-
form plate temperature will introduce error into the results.
There-
fore, each data run must start from the most uniform temperature
possible.
The heating arrangement consisted of a temperature-controlled hot
plate with a cast aluminum top approximately 15 cm (6 in) square which
was inverted in the middle of the aluminum plate as shown in Figure 8.
The perimeter of the aluminum plate which was not in direct contact
with the hot plate surface was insulated with newsprint during the
warm-up period to reduce the heat loss and allow a more uniform plate
temperature to develop.
The center plate temperature was held at
approximately 93°C (200°F) while the outer perimeter temperature was
24
Perimeter of
Convecting surface
insulated during
warm up period
Temperature controlled
electric resistance
hot plate inverted
on convecting surface
Insulation Block
Assembly
METHOD USED TO WARM ALUMINUM PLATE
FIGURE 8
25
monitored with the data logger.
Once the outer perimeter reached equi-
librium with the inner plate temperature, the hot plate was removed
and the entire insulation block assembly was covered with a 58 x 58 x
5 cm block of polyurethane foam insulation.
The plate was allowed to
cool to approximately 77°C (170°F) to further stabilize and assure a
uniform plate temperature.
A maximum deviation of 2°C between plate
center and plate edge was allowed with a common deviation of less
than 1°C.
Once a stable plate temperature of 77°C had been reached, the heat
transfer surface was placed in the air stream which was flowing from
the wind tunnel at a predetermined velocity.
The desired angle of im-
pingement was set with the tripod mount and the data logger was programmed to read the temperature of the plate and the corresponding time
at 15 to 60-second intervals depending on the rate of temperature decay.
The temperature of the plate was monitored between 77° and 27°C, although the temperature region between 77° and 49°C was not used for
data purposes due to the inconstant apparatus conductance as was determined in Appendix C.
Plate temperatures below 35°C were avoided
due to measurement inaccuracies as was determined in Appendix B.
2.3.2
Measurement of Temperature and Time
The temperature and time measurements were made with an Esterline
Angus Model PD 2064 programmable data logger.
The data logger accepts
thermocouple signals as an input and provides the temperatures and
associated times as output.
Both visual and hard copy output were
26
provided.
The operation procedure consisted of defining a start time
and the interval between measurements.
vided automatically.
All other functions are pro-
The final output is a very accurate profile of
temperature of the plate and the associated times.
Data reduction requires the time for the temperature to decrease
from 49°to 38°C and the specific conditions of the test.
Figure 9
illustrates a typical time/temperature profile along with the associated upper and lower temperature limits used to bracket the region of
data collection.
2.3.3
Measurement of Air Velocity
Measurements of the
air velocity were made with a pitot tube
attached to a precision micromanometer, and cross-checked with a 10 cm
diameter averaging vane-type anemometer.
The velocity was measured
2.5 cm (1 in) above and parallel with the leading edge of the aluminum
plate, while the insulation block was horizontal and touching the
bottom of the exit nozzle.
This was outside the boundary layer and
was thought to be a good representation of the free stream velocity.
Velocity readings were taken across the plate width and averaged.
The
same method was used to determine the exit velocity for all orientations of the plate relative to the air flow.
Pitot tube velocity
measurements of the air stream velocity were performed with the insulation block oriented at each angle of attack and compared to the
horizontal measurements to assure that no blockage of the exit port
was measurable.
The measurement uncertainty ranged
from
±.1 to
27
78
(170)
\
a)
'S
4C-CS.'
54
-
(130)
a)
,
F
4s1t3_ a
CUE
t
a)
all_ 1 e.rops.r.a 1
u.r...e._
____
.
43
7_ (110)
\
32
\--
N
(90)
''
.--.
21
(70)
0
5
10
15
Time in minutes
TYPICAL TIME/TEMPERATURE PROFILE
FIGURE 9
20
28
±.3 ft/sec, with the greatest uncertainty encountered at lower velocities.
The uncertainty of the velocity measurement as a function of
velocity is shown in Appendix E.
29
2.4
DATA REDUCTION
The method of analysis used requires the assumption of a uniform temperature throughout the plate.
The association between the
accuracy of this assumption and the Biot number was explained in
Section 2.1.
Due to the low value of the Biot number for the system
employed, the variation of temperature throughout the plate should
be small.
The data taken demonstrated that the plate was not at the
same temperature everywhere, but tended to be cooler at the leading
edge and warmer at the trailing edge.
The temperature was measured
2.5 cm from the edge and at the centerline of the plate.
The data
showed an overall plate deviation of approximately 2% at the highest convection rates with smaller variations for lower convection
rates.
The temperature at the geometric center of the plate was
taken as the most representative of the overall temperature in all
subsequent data reduction.
2.4.1
Mathematical Model and Analysis
The nature of the system involved can be viewed as a simple
energy balance, in which the internal energy change of the body is
equal to the energy lost to the environment.
Consider an arbitrary
volume represented in Figure 10 where the losses due to radiation
are ignored because it accounts for no more than 0.5% of the total
energy loss.
30
The internal energy change per unit time is represented by equation
(2-5).
pvc
dT
internal energy change
(2-5)
TIT
The convection and conduction losses can be represented by equation
(2-6),
(H + K(T))(T - T.) = energy losses
(2-6)
where K(T) represents the temperature-dependent conduction loss through
the insulation and H represents the convection loss from the exposed
surface.
If K(T) can be approximated by a constant over a given tem-
perature range, then K(T) can be replaced by K which merely represents
31
the total energy loss due to conduction through the insulation.
Equating equation (2-5) and equation (2-6) results in the differential equation (2-7), which describes the energy balance for the
system.
atpvc
= (H + K) (T - Ted
dt
(2-7)
Equation (2-7) can be rearranged to the form of equation (2-8)
(H+K)dt
pvc
dT
(T-Tm)
(2-8)
Integrating both sides of equation (2-8) and evaluating the arbitrary
integration constant with the boundary condition that the temperature
at some arbitrary start time, t = 0, is equal to a constant T., results
in equations (2-9) and (2-10).
Substitution of the constant determined
in equation (2-10) into equation (2-9) yields equation (2-11).
r
(H+K)
0-7c t "r ul
,
tn(T-T.) =
@
t = 0, C1= 9n (To - T.)
bl(T- T
)
=
(H+K)
t + Ln(To - T.)
pvc
(2-9)
(2-10)
(2-11)
Equation (2-11) can be rearranged into a more useful form which is
shown in equation (2-12).
The convection heat loss represented by H
can be replaced by the average convection film coefficient times the
32
exposed surface area giving equation (2-12)
pvc
-
IT-Too
kn=
to
1
K
To-Ted
A
(2-12)
The resulting convection film coefficient can be nondimensionalized by
multiplying by the plate length L and dividing by the thermal conductivity of the moving air stream K'.
This yields the average Nusselt
number as shown in equation (2-13).
1
K1
1'71j T.,-Toj
Al
mc
"u
Tit
L
K'
171
to
[ T-To,
(2-13)
The results can also be expressed in terms of the Colburn j-factor as
defined for heat transfer by equation (2-14)
Nu
j = StPr2/3
(2-14)
RePrl /3
Using the appropriate substitutions for the Reynolds, Prandtl and
Nusselt numbers results in equation (2-15), which was used for the
reduction of the data collected.
Pr
I
2/3
p c
I
[
me
to
I T -T.
1
To-T. j
K
(2-15)
j
The Reynolds number which characterizes the flow was evaluated at the
length of the aluminum plate, 0.3048 m (1 ft), for all cases.
33
The air velocity ranged from 1.7 to 7.6 m/s (5.6 to 25 ft/sec)
which corresponded to a Reynolds number range of 31,900 to 140,000.
Table 2 lists the values for all the parameters used to evaluate the
appropriate j-factor in equation (2-15).
2.4.2
Experimental Data
Each data run resulted in a recorded temperature decay profile
under constant ambient conditions from 77° to 27°C (170° to 80°F).
Greatest accuracy was achieved by picking the desired start temperature
in the region of 49°C (120°F) and a stop temperature in the region of
38°C (100°F).
The time associated with these two temperature measure-
ments is the time required for the aluminum plate to decrease in temperature from approximately 49° to 38°C (120° to 100°F).
Appendix D
lists the start, stop, and ambient temperature; with the associated
decay time and the calculated j-factors for the orientation and wind
velocity listed.
The data were recorded in °F, feet per second, and
minutes for temperature, wind velocity, and time, respectively.
2.4.3
Uncertainty Analysis
An uncertainty analysis was performed on equation (2-16) with the
worst case uncertainty of the various parameters as stated in Table 2.
The temperature and air velocity measurements proved to be the largest
sources of error, indicating a maximum j-factor uncertainty of .00024,
which corresponds to a maximum uncertainty of 4.5%.
34
TABLE 2
VALUES OF PARAMETERS USED FOR DATA
REDUCTION WITH EQUATION (2-15)
Parameter
Name
Pr
Prandtl #
.7055
P'
density of air
.07226
Uncertainty
Units
Value
± 0.0015
-
lbm/ft
± 0.00005
3
± 0.5
174.5
P
density of aluminum
c
specific heat of
aluminum
.225
Btu/lbm°F
± 0.0005
c'
specific heat of air
.2403
Btu/lbm°F
± 0.0001
u.
free stream air
velocity
measured
± 0.3 max
ft/sec
2.288
ibm
measured
minutes
± 0.002
m
mass of aluminum
plate
t
time
A
area of aluminum
plate
To
temperature
measured
°F
±
.05
T.
start temperature
measured
°F
±
.05
T
ambient temperature
measured
°F
±
.05
K
thermal conduction
loss (measured)
Btu
±
.001
ft
1
0.1030
± 0.00016
2
negligible
hr °F
_
.
35
2.5
RESULTS AND COMPARISON WITH PREVIOUS WORK
The convection film coefficient, as reflected to the Colburn jfactor is dependent upon the type of flow exhibited in the boundary
layer.
A change in the boundary layer from laminar to turbulent is
characterized by a decrease in slope in the j-factor curve.
Accord-
ing to Kays [5], the j-factor associated with laminar flow will be
dependent upon the Reynolds number to the -0.5 power, while that
associated with turbulent flow will be dependent upon the Reynolds
number to the -0.2 power.
Transition flows will probably have a
slope which is bounded by these two values.
Figure 11 shows the results which were provided by Jurges [1]
and associated calculated flat plate curves representing a laminar
and a turbulent boundary layer based on correlations presented by
Kays [5].
Both flat plate curves have been corrected to account for
the effects of approximately 0.35 m of unheated starting length
associated with Jurges experimental apparatus.
The j-factor for
laminar flow including the unheated starting length is shown in equation (2-16).
in Kays.
This was derived from Pohlhausen solution described
The contribution due to the unheated starting length in-
creased the j-factor by approximately 47%.
(2-16)
j = .976 Re-1/2
Equation (2-17) derived from Kays' work
describes the j-factor
associated with a turbulent boundary layer which was increased by
approximately 10% due to the unheated starting length.
0.007
,
Jurges
..,,
0.006
* Flat Plate-''
0.005
,,,, ..
* Flat Plate
0.004
,,, ,,,,,,,,
,,,,,
.
,,,,,
Turbulent
J
444,,,,,,,
,,,,,,
'44.1%%
0.003
*corrected for unheated starting length
0,002
20,000
30,000
40,000
60,000
80,000
Reynolds Number
100,000
RESULTS OF PREVIOUS ZERO DEGREE ANGLE OF ATTACK STUDIES
Figure 11
150,000
200,000
37
(2-17)
j = .0368 Re-0'2
The relationship between Jurges results and the calculated laminar and turbulent flow predictions shown in Figure 11 seem to indicate that the flow experienced by Jurges was laminar at the lower
Reynolds numbers and became progressively more turbulent at higher
Reynolds numbers.
The larger values for the j-factors in Jurges work
could be attributed to unaccounted thermal losses or consistent measurement error.
Figure 12 compares the Jurges results and the zero-degree angle
of attack data measured by the author.
The same approach used to
develop equation (2-17) was used to develop a turbulent flow j-factor
equation which takes into account the effect of the unheated starting
length imposed by the insulated border in the present study.
This
is shown in equation (2-18) and is plotted in Figure 12 for
comparison.
(2-18)
j = .0351 Re-0'2
The slope of the best fit line through the data is evidence of a
high degree of turbulence, and was probably due to the effect of the
leading corner of the insulation block which protruded approximately
1.2 cm into the air stream.
This arrangement was used to minimize
any blockage of the air stream over the plate due to tubular flow
straighteners in the exhaust port nozzle.
Transition from a laminar
to turbulent boundary layer due to steps in flow path has been confirmed by Sam, Lessmann, and Test [4].
0.007
Jurges
0.006
*Flat Plate
0.005
.... ,
/Turbulent
.,,,,,,,,,
_
.,,
0.004
,........_
--
......
--
.
, ,. ,,,,
-..........
....
0.003
-,
,
-........,,..,,
*corrected for unheated starting length
0.002
20,000
0,000
40,000
60,000
80,000
Reynolds Number
100 000
150,000
200,000
ZERO DEGREE ANGLE OF ATTACK DATA
CO
Figure 12
39
The data which are plotted in Figure 13 for a 30-degree angle
of attack indicate a nearly laminar flow.
The best fit line does
show a small tendency to rotate towards a slope which indicates
turbulence was present.
(Turbulence will tend to lower the j-factor
at lower Reynolds numbers and raise it at higher Reynolds numbers for
the range of 30-degree data.)
The actual turbulence intensity of
The
the wind tunnel was not measured and probably is fairly large.
tendency toward laminar flow confirms results of Sam [4] which indicate that angles of attack above 30° tend to stabilize the flow.
The
disagreement evident at lower Reynolds numbers between the 30° degree
data and the Pohlhausen solution shown in equation (2-19) and plotted
in Figure 13 is probably due to turbulence.
= .876 Re-1/2
(2-19)
It is the author's opinion that the Pohlhausen flat plate solution,
which has been corrected for unheated starting length, represents the
j-factor associated with a 30-degree angle of attack for the present
data.
The 45-degree angle of attack data plotted in Figure 14 appears
to be the result of a turbulent boundary layer, as evidenced by the
slope of the j-factor curve.
According to Sam [4] the increasing
angle of attack should continue to reinforce the laminar nature of
the flow.
The increased turbulence compared with the 30-degree angle
of attack data indicates that a possible interaction between wind
tunnel turbulence and angle of attack existed in the region of
0.007
\ \ \ \ \ \
\ \ \ \ \
\ \\
0.006
%\\*\
Sparrow
\*\
8'
0.005
\\*\\
*F-1-a
t P teLamin a
0.004
.....
r
- --:*-,-......
3
\N\*\\
\\
0.003
*\u,..
.\\t,4,,,;,,,,,,,
*corrected for unheated starti
length
0.002
20,000
30,000
40,000
80,000
60,000
Reynolds Number
100,000
30-DEGREE ANGLE OF ATTACK DATA
Figure 13
0.007
%4k
0.006
0x,
A%\,,
A\`'\v
\ \ \\
0.005
0.004
Spa-row & Tien
--\\\\\
\\\\\\\\
.*
0004
Flat Plate
Turbulent
h
%%%%%%%%% \\1*-
3
%N%.,
''"k...
-.
%%%%%%%
.--,.......
0.003
'\\
0.002
20,000
30,000
40,000
60,000
80,000
Reynolds Number
100,000
45-DEGREE ANGLE OF ATTACK DATA
Figure 14
150,000
200,000
42
45 degrees, which overpowered the stabilizing effects of the larger attack angle.
Because Sparrow and Tien's results exhibit a laminar flow,
the turbulence data taken at 45 degrees is not directly comparable.
Figures 15 and 16 provide the data which are the most closely related to the work of Sparrow and Tien.
The slope of the data indi-
cates that a very nearly laminar flow situation existed.
This con-
firms the predictions of Sam [4] concerning the stabilizing of laminar flow at high angles of attack.
The data provided in Figure 16 for
a 90-degree angle of attack can be directly compared to a laminar
boundary layer wedge problem in which u. = cxm as described in Kays
[5].
In this situation, the j-factor equation is derived to be
equation (2-20).
(2-20)
j = 1.114 Re-1/2
Equation (2-19) is plotted in Figure 16 and indicates good agreement
with the experimental results determined here.
Figures 15 and 16 indicate a dependence of j-factor upon angle
of attack which was not observed by Sparrow and Tien.
It is the
author's opinion that these discrepancies can be attributed to the
different flow geometries which existed in the experimental arrangement.
Sparrow and Tien measured the j-factor for a 7.6 cm square
plate suspended in a large low-turbulence wind tunnel.
Thus,
the flow by the test specimen was relatively unrestricted.
In con-
trast, the system employed here consisted of a convecting surface
30.48 cm square surrounded by a 14 cm border, located external to a
0.007
0.006
li
\\
\\ \\
\ \ \ \ \ ...
0.005
.
---,
.
..,
Sparrow & Tien
---1-
//
0.004
..
..
..
,
.
.
--
..
.,
.,
3
.
/
.
..
...,.
*/
0.003
...
yyiyyy/
0.002
20,000
30,000
40,000
60,000
80,000
Reynolds Number
100,000
60-DEGREE ANGLE OF ATTACK DATA
Figure 15
150,000
200,000
1111110111
11111111111111111111111111
111111111111
,1111
Nunn
sczs
-v\e\
11111111111111111111101111111
10111
011001
1111110=
11111111111M
O
o
45
43 x 25 cm wind tunnel exhaust port.
Because the total dimensions
of the insulation block exceeded the width of the wind tunnel exit
port, largely two-dimensional flow around the test surface was
probable at all angles of attack except 90 degrees.
Figure 17 summarizes the results for each angle of attack from
zero to 90 degrees.
The curves which represent zero and 45 degrees
appear to exhibit turbulence due to their slope and cannot be directto the laminar flow work of Sparrow and Tien.
ly compared
The mea-
sured difference between the zero and 45-degree data is small and
could, within experimental error, indicate an independence of the
turbulent flow j-factor upon angle of attack (between zero and 45
degrees only).
The 30-degree data do not substantiate this conclu-
sion because the flow experienced at that angle appeared to be
laminar.
The data shown in Figure 17 corresponding to angles of attack
of 30, 60 and 90 degrees, which exhibited laminar boundary layer flow,
appear to demonstrate a general upward shift of the j-factor curve due
to increasing angle of attack.
2.5.1
Applicability to Solar Collectors
The convecting surface surrounded by an insulating border is
similar to the situation encountered when solar collectors are mounted
on a roof.
The results of the present work seem to indicate that
when the boundary layer is laminar, the convection film coefficient
for an angle of attack of 90° will be 1.3 times the coefficient for
0.007
0.006
90 Degree
0.005
60 Degree
z-.z-
---,-----,..-
-,-,
-,
---
--
.
---_
---__
--
0.004
II
0.003
Oil
1111r,
ill
ere Degree
111q11111
45 Deg-ee
ill30
0.002
20,000
30,000
40,000
80,000
60,000
Reynolds Number
D
gre
100,000
MEASURED J-FACTOR DEPENDENCE UPON ANGLE OF ATTACK
Figure 17
150,000
200,000
47
an angle of attack of 30°.
This can be applied to a typical solar
collector, as Sparrow [2] has done, which is 2.4 m (8 ft) square and
exposed to a wind velocity of 3 m/s (10 ft/sec).
The Jurges equation
2
would predict a convection film coefficient of about 17.5 w/m °C
(3.1 Btu/hr ft2°F) while the equation proposed by Sparrow would
2
yield a value of 6.15w/m2 °C (1.1 Btu/hr ft °F).
The present work
yields a value of 5.78 w/m2°C (1.02 Btu/hr ft2°F) for an angle of at2
2
tack of 30 degrees and a value of 7.34 w/m °C (1.29 Btu/hr ft °F) for
an angle of attack of 90 degrees.
The results also suggest that the existence of a turbulent boun-
dary layer can increase the convection film coefficient in the above
example to 11.08 win°C (1.95 Btu/hr ft2°F), or 1.9 times the laminar
boundary layer coefficient for an angle of attack of 30 degrees.
The
present data, due to the slope of the j-factor curve, seems to indicate
increasingly laminar flow at larger angles of attack, which confirms
the findings of Sam [4].
This implies that steeper angles of attack
can retard the development of a turbulent boundary layer and the associated higher convection film coefficients.
It is important to note that the Reynolds number associated with
the example in this section was 460,000 while the highest Reynolds
number data measured in this investigation was 140,000.
Reynolds number measured by Sparrow and Tien was 100,000.
sary extrapolation of the data should be kept in mind.
The highest
The neces-
48
SECTION III
EXTENDING THE CONVECTION FILM COEFFICIENT DETERMINED FOR
CONSTANT VELOCITIES TO RANDOM VELOCITY DISTRIBUTIONS,
AN APPROXIMATE ANALYTICAL APPROACH
49
An accurate prediction of the convection film coefficient which
a real system experiences is dependent on three factors.
The first
consideration is an accurate prediction of the dependence of the convection film coefficient on velocity of the wind, which was investigated in Section II.
The second variable expresses the effect of the
random variations of the velocity
on the prediction of a time-
averaged convection film coefficient, which will be considered in
this section.
The final consideration takes into account the random
directional orientation of the wind relative to the heat transfer surface.
More data are required in this area before accurate predictions
can be made.
The geometry of the heat transfer surface and all sur-
rounding objects can have an effect on the ultimate directional orientation of the wind.
This problem becomes immensely complex and is
left to future study.
3.1
EVALUATING THE CONVECTION FILM COEFFICIENT USING
THE AVERAGE AND THE STANDARD DEVIATION
OF THE WIND VELOCITY
The nature of the predicting equation for the convection film coefficient demonstrates a nonlinear dependence on wind velocity.
If the
convection film coefficient were dependent on the velocity in a strictly linear manner, then the average ambient wind velocity could be used
to calculate the time-averaged convection film coefficient exactly.
The rate at which the convection film coefficient increases with wind
velocity
decreases at higher and higher velocities.
Consequently,
simply using the average ambient wind speed in predicting the time-
50
averaged convection film coefficient will result in an error by not
taking the distribution of wind velocity into account.
The goal of this section is to predict the time-averaged convection film coefficient hw exactly with only V and a known for the
velocity distribution.
It is important to note that "exact only means
that no error will be introduced due to the random nature of the wind
velocity distribution.
The method developed requires a convection
film coefficient predicting equation in the form of a second order
This polynomial would
polynomial over a fixed range of velocities.
appear as equation (3-1).
hw(t) = a. +
Where h
w
+ a2V(t)
2
(3-1)
represents the convection film coefficient due to the ambi-
ent wind, V(t) is the ambient wind velocity as a function of time,
and the coefficients a,, al, and a2 are known constants.
The time-
averaged convection film coefficient is defined to be equation (3-2).
Substitution of equation (3-1) into equation (3-2) results in equation (3-3).
jr
hw =
hwmdt
(3-2)
Fig = t jr (a0 + W.1(0 + a2V(t)2)dt
(3-3)
Carrying out the integration indicated in equation (3-3) using
integration by parts yields equation (3-4) .
hw =
1
jr a.dt +
1
TfaiV(t)dt
+
1
2
"la V(t) dt
I
2
(3-4)
51
Bringing the constants through the integral sign yields equation (3-5).
1
w
1
V(t)dt
(t)dt + az
aol f dt + alt
=
t
f V(t) 2dt
(3-5)
Equation (3-5) represents thenexaceconvection film coefficient as a
function of the velocity.
The exact velocity distribution with re-
spect to time must be known to evaluate equation (3-5) directly.
The
requirement of knowing the true velocity distribution can be replaced
by knowing the average and standard deviation of the velocity distribution.
Several definitions are required to perform the transformaThe time-averaged velocity can be expressed by equation (3-6).
tion.
V =
(3-6)
tf V(t)dt
The statistical variance of the velocity is defined by equation
(3-7).
2
S
1
=
t
f (V(t)
(3-7)
- V)2dt
Where the square root of the variance is defined to be the standard
deviation a, as in equation (3-8).
a = iN/ST
(3-8)
Inspection of equation (3-5) reveals that the second integral on the
RHS is simply the time-averaged wind velocity V, equation (3-6), and
the first integral equals one.
Carrying out the indicated simplifi-
cations and placing brackets around the remaining integral yields
equation (3-9).
52
= a, +
aiV+
a2lf V(t)2dt
(3-9)
The remaining integral in equation (3-9) can be simplified by expanding equation (3-7).
This expansion is performed in two steps shown
in equation (3-10) and (3-11).
1
-f
S2
S
2
=
Jr
1 jr
(v(t) - v) 2
(V(t)
2
(3-10)
-2
- 2VV(t) + V )dt
(3-11)
Carrying out the indicated integration through equation (3-11) yields
equation (3-12).
S2
= t jrv(t)2dt
r 2VV(t)dt +
t j
f V2dt
(3-12)
The last two terms on the RHS can be simplified by noting that V is a
constant.
The resulting simplification yields equation (3-13).
2V1jr V(t)dt + Vitfdt
S2 = -r-lfV(t)2dt -
(3-13)
The second integral on the. RHS of equation (3-13) is simply V while
The resulting simplification yields equa-
the third integral is one.
tion (3-14).
s2
f V(t)2dt
-
2V2 + V2
(3-14)
Rearranging of equation (3-14) yields equation (3-15).
71t-fV(t)2dt = S2 +
1/2
(3-15)
Equation (3-15) defines the remaining integral in equation (3-9).
53
Substitution of equation (3-15) into equation (3-9) will yield equation (3-16).
(3-16)
= a. + aiV + a2(S2 + V2)
It is obvious from the form of equation (3-16) that the error
introduced when evaluating a second order approximation of the timeaveraged convection film coefficient by simply using the average
velocity is equal to the magnitude of the coefficient a2 times the
variance of the velocity distribution.
Equation (3-16) can also be
rewritten in the form of equation (3-17).
w
I
V
=
w
1 + a2S2
(.3 -17)
V
The magnifitude of a2 in equation (3-1) governs the degree of error
introduced by using the average wind velocity in calculating the convection film coefficient.
This was demonstrated in equation (3-17),
since the convection film coefficient calculated at'the average
velocity V will be equal to the general convection film coefficient
when the velocity is constant or 17,4 is linear.
An indication of the
magnitude of the error introduced by using the average wind velocity
can be achieved by forming a second order interpolating polynomial
for the appropriate convection film coefficient predicting equation,
and applying the derived equations to a physically measured velocity
distribution with a calculated average velocity and standard deviation.
Comparison of the two results will yield the error which is
attributed to the use of the average velocity approximation.
This
54
can be demonstrated by considering the convection film coefficient
predicting equation which was determined by Sparrow and Tien [2]
Combining equation (3-13) and the
and is shown in equation (3-18).
definition of the j-factor for heat transfer shown in equation (2-1),
along with the associated definitions of the Reynolds number, Stanton
number, and Nusselt number shown in equations (3-19), (3-20) and
(3-21), respectively, will yield a relationships for the average convection film coefficient which is shown in equation (3-22).
The
parameters used in equation (2-1) and equation (3-18) through (3-21)
are defined in Appendix A.
j = .931 Re
j = StPr
-1/2
2/3
=
(3-18)
Nu
1/3
(2-1)
RePr
(3-19)
Re =
St =
Nu
(3-20)
RePr
(3-21)
Nu =
=
144,01/2Pr1/3
.931
(3-22)
1/2
1/2
L
Table 3 shows the values used for the various parameters to arrive at
2
2
equation (3-23) expressed in w/m °K (Btu/hr ft °F) with the velocity
expressed in m/s (ft/sec)
=
5.460/2 (IT
=
.9633V1/2)
(3-23)
55
TABLE 3
VALUES USED FOR PARAMETERS IN EQUATION (3-22)
Value
Parameter
3.2
L
.3048 m (1 ft)
Pr
.706
K
0.0268 w/m°K (0.0155 Btu/hr ft°F)
v
1.653 x 10
-5
2
m /S (17.79 x 10
-5
2
ft /sec)
EXAMINATION OF THE ERROR INTRODUCED BY USING THE AVERAGE
WIND VELOCITY TO PREDICT THE AVERAGE
CONVECTION FILM COEFFICIENT
Equation (3-23) must be expressed in the form of a second order
polynomial with constant coefficients before the results of equation
(3-16) can be applied.
The conversion can be accomplished in many
ways, but one which is most applicable is the construction of a
second order interpolating polynomial through three selected points.
Three points of interest to this investigation might be the convection
film coefficient observed for velocities of 0.6, 7.6, and 15.2 m/S
(2, 25, and 50 ft/sec.).
Table 4 lists the velocities and the asso-
ciated convection film coefficients as calculated by equation (3-13)
which results from applying the parameters shown in Table 3 to
56
The interpolating polynomial I(V) can be found by
equation (3-22).
placing a second order Lagrangian Polynomial through the three points
listed in Table 4.
TABLE 4
SELECTED POINTS
Velocity m/s
(ft/sec)
hw w/m
2°C
4.23
(Btu/hr ft
2°F)
(1.36)
0.6
(2 )
7.6
(25)
15.0
(4.81)
15.2
(50)
21.3
(6.81)
The steps involved in this polynomial generation are shown in equations (3-24), (3-25), and (3-26).
I(V) = 1.36
( V-25)(V-50)
(2-2g)(2-50)
+ 4 8
(25-2 ) (25-5d)
+
6.81(2:2 (.502 )
(3-24)
I(V) = .001232 (V2 - 75V + 1250) - .008365 (V2 - 52V + 100)
(3-25)
+ .005675 (V2 - 271, + 50)
(3-26)
I(V) = .98725 + .18936V - .001458V2
Equation (3-26) is one of many second order polynomials which approxi-
mate Sparrows' and Tiens' convection film coefficient predicting
equation between 2 and 50 ft/sec.
Figure 18 shows the degree of
correspondence between the approximate polynomial and the original
equation.
Therefore, in the region of velocities from 2 to 50 ft/sec,
57
34. )
(6.0 )
28.
../
3
[
(5.0
L)
`'41E
22.
1
3?(4.0)
.
/
//
/_,__//
/
/
/
)7
./(
iti
5.46 V 1/2
.96725 + .18936V - .0)1458V
2
/
//e///./
4
5.7
(1.0)
)
(10) 3.0
(20)6.1
(30)9.1
Velocity (ft/sec)
(40)12.2
m/s
AGREEMENT BETWEEN INTERPOLATING POLYNOMIAL
AND GOVERNING EQUATION (3-23)
Figure 18
(cn)
15.2
58
the convection film coefficient as predicted by Sparrow [2] can be
approximated by the second order polynomial I(V), as shown in equation
(3-27).
F
(3-27)
= .98725 + .18936V - .001458V2
The error introduced by using the average wind velocity to calculate
the convection film coefficient may be evaluated by applying the results
of equation (3-16) to equation (3-27).
Necessary wind velocity data
were recorded by the reactor building monitor at Oregon State University on November 29 and 30, 1978, and is tabulated in Appendix E.
The
convection film coefficient was evaluated for two separate cases.
Case 1
used the average wind velocity to predict the convection film
coefficient for November 29 and 30, 1978 as shown in equation (3-28).
Case 2 used the average and standard deviation of the wind velocity on
November 29 and 30 to predict the convection film coefficient using
the results of equation (3-16) as shown in equation (3-29).
(3-28)
.001458V2
Case 1
71
=
.98725 + .18936V
Case 2
T1
=
.98725 + .18936V - .001458 (S2 + V2)
-
(3-29)
The results of this illustration are shown in Table 5.
TABLE 5
ERRORS INTRODUCED BY IGNORING STATISTICAL INFORMATION
Calculated h
Btu/hr ft2°F
Velocity
Data ft/sec
V
a
Case 1
Case 2
14.23
8.90
3.386
3.271
% change
Date
11/29-30/78
3.4%
59
This example shows that the dependence of the convection film
coefficient on the variation in velocity is relatively small for this
particular situation.
However, it also shows that due to the decreas-
ing nature of the convection film coefficient prediction equation,
the use of an average velocity will always over-predict the average
heat loss.
60
SECTION IV
CONCLUSIONS
61
CONCLUSIONS
4.1
DEPENDENCE OF THE CONVECTION FILM COEFFICIENT
UPON ANGLE OF ATTACK
The investigation observed flows ranging from laminar, as the
slope of the j-factor curves seemed to indicate, at angles of attack
of 30, 60, and 90 degrees, to turbulent, as the slope of the j-factor
curve seemed to indicate, at angles of attack of zero and 45 degrees.
The describing equation for laminar flow of 30 degrees was found
to be represented by equation (4-1).
(4-1)
j = 0.876 Re-1/2
The describing equation for laminar flow at 90 degrees was thought to
be represented by equation (4-2).
(4-2)
j = 1.114 Re-1/2
The describing equation
for laminar flow of 60 degrees should have a
multiplier which lies between 0.876 and 1.114 and will probably be
closer to the higher value.
The turbulent flow which was observed
at angles of attack of zero and 45 degrees resulted in a j-factor
curve described by equation (4-3) for both cases.
j = .0335 Re
-0.2
(4-3)
Equations (4-1) and (4-2) indicate a measured dependence of the
62
convection film coefficient upon angle of attack which is beyond the
expected experimental error.
These results are in disagreement with
the previous work of Sparrow and Tien [2][3] which indicate that the
equation j = 0.931 Re
-1/2
should apply for all angles of attack.
It
is the author's opinion that the disagreement is probably due to the
differences in the air flow by the test surface.
Sparrow and Tien used
a 7.6 cm (3 in) square test surface inside a 30 x 60 cm (12 x 24 in)
wind tunnel.
In contrast, the present study used a 58.4 cm square
surface in front of a 25.4 x 43.2 cm (10 x 17 in) wind tunnel exhaust
nozzle.
The flow experienced by Sparrow and Tien was three-dimensional
while the flow in the present study was more or less two-dimensional.
The results at a 90-degree angle of attack agree with previous
work reported in Kays [5] for the corresponding laminar boundary layer
wedge flow.
The results also seemed to indicate that steeper angles of attack
increased the tendency toward a laminar boundary layer.
with recent findings by Sam, Lessmann, and Test [4].
This agrees
63
4.2
RANDOM WIND VELOCITIES AND THE AVERAGE
CONVECTION FILM COEFFICIENT
The results indicate that if statistical knowledge of the wind
velocity distribution in the form of an average velocity
and a standard
deviatioh a were available, then a second order constant coefficient poly-
nomial approximation of the convection film coefficient defining
equation can yield results which account for the random nature of
Equation (4-4) describes the form of the poly-
the wind velocity.
nomial expression for the convection film coefficient, and equation
(4-5) describes the interaction of the statistical information which
accounts for the random nature of the wind velocity.
h
h
w
w
= a. + a1V(t) + a2V(t)
= a. +
2
I,
+ a2(V2 + a
2
)
(4-4)
(4-5)
From the form of equation (4-5), it is obvious that the magnitude of
the error encountered by using the average velocity only in equation
(4-4) will be a2a
2
.
Since the coefficient a2 is always negative in
a convection film coefficient predicting equation, the error encountered will tend to overpredict the convection film coefficient.
How-
ever, the magnitude of the error was found to be less than 4% in the
example provided.
Therefore, the random nature of the wind velocity
can probably be ignored in most cases.
The effect of random wind directions was not considered in
the present investigation.
The variation of the convection film
64
coefficient with angle of attack, which was observed in Section II,
indicates that changes in directional orientation of the wind could
affect the total average convection coefficient.
This is because a
change in directional orientation of the wind effectively changes
the angle of attack relative to the stationary structure.
65
4.3
FUTURE RESEARCH
The results of the two-dimensional heat transfer studies currently
under way, which the work of Sam, Lessmann, and Test [4] has contributed to, should provide information which is directly applicable to
the results of this report.
Future research should utilize a thermal decay method, such
as the one employed here, in a three-dimensional flow such as was used
by Sparrow and Tien [2][3].
This should confirm the author's opinion
that the results obtained here which disagree with the results of
Sparrow are the consequences of the differences between the two- and
three-dimensional flows.
66
BIBLIOGRAPHY
1.
Dr. Ing-Walter ()Urges, Der nrmeUbergang an einer ebenen Wand,
Biehefte Zum Gesundheits-Ingenieur, Reihe 1, Beiheft 19,
November 1924.
2.
E. M. Sparrow and K. K. Tien, Forced Convection on an inclined
and yawed flat plate - Application to Solar collectors, J.
Heat Transfer 99, 507-512 (1977).
3.
K. K. Tien and E. M. Sparrow, Local heat transfer and fluid flow
characteristics for airflow oblique or normal to a square
plate, Int. J. Heat Mass Transfer, Vol. 22, No. 3-A, pp. 349360
4.
1979).
R. G. Sam, R. C. Lessmann, and F. L. Test, An experimental study
of flow over a Rectangular Body, Journal of Fluids Engineering,
Transactions of the ASME, Volume 101, December 1979, pp. 443448.
5.
W. M. Kays, Convective Heat and Mass Transfer, New York: McGrawHill Book Company, 1966, pp. 203-243.
67
APPENDICES
68
APPENDIX A
NOMENCLATURE
A
surface area of the convecting surface
a
arbitrary constant (with subscripts)
H
total thermal convection rate
h
local convection film coefficient
h
w
convection film coefficient due to wind
average convection film coefficient
time average convection film coefficient due to wind j-factor
K
total thermal conductance loss
K'
thermal conductivity of the air
k
thermal conductivity
k'
mass transfer coefficient
L
length
t
characteristic length
m
mass of the aluminum plate used in apparatus
m
mass transfer rate
Nu
Nusselt number
Pr
Prandtl number
Sc
Schmidt number
St
Stanton number
S
2
Statistical variance
T
temperature
Te
start temperature
69
Tm
ambient temperature
t
time
free stream wind velocity
V
average velocity of wind throughout time
V(t)
velocity of wind as a function of time
volume of the aluminum plate used in apparatus
kinematic viscosity of air
p
density of aluminum
p'
density of air
pnw
wall napthalene vapor concentration
prim
free stream napthalene vapor concentration
standard deviation
70
APPENDIX B
DETERMINATION OF STABLE TEMPERATURE DECAY REGION
(Sphere Tests)
During the course of this investigation, a dependence of the
convection film coefficient upon plate temperature was observed.
This caused concern for the validity of the techniques employed.
An
investigation determined the temperature region over which valid data
could be obtained.
The measured temperature dependence of the flat plate convection film coefficient is shown in Figure 19.
is broken into three temperature regions.
The temperature scale
A plate temperature above
49°C exhibits a steeper slope mainly due to increasing insulation
losses at higher temperatures.
The slope of the curve in the region
of 49 to 38°C is mainly due to temperature dependent properties of
the aluminum plate and the air.
The sharply decreasing slope between
38°C and ambient is thought to be due to the increasingly invalid
assumption of a uniform body temperature and conduction resistance
between the thermocouple and the body.
The existence of the lower temperature boundary was confirmed by
running identical tests on a 2.54 cm (1
in) diameter copper sphere
suspended in the wind tunnel air stream by a thermocouple which was
soldered in a hole drilled into the geometric center of the sphere.
This arrangement eliminated contributions due to insulation losses.
Figure 20 shows a graphical representation of the measured convection
71
34.0
(6)
28.4
(5)
22.7
0,1E (4)
Increasin ) insulation
1
1..'=
DSS
Vow
Measurement
(Measurement
errors
11.3
(2)
5.7
(1)
0
(175)79
(100)38
Plate Temperature(°F) °C
(150)66
(125)52
MEASURED TEMPERATURE DEPENDENCE OF THE
FLAT PLATE CONVECTION FILM COEFFICIENT
Figure 19
(75)24
72
51.0
(9)
45.4
(8)
(..)39.7
NE
)
3: 34.0
c (6)
+- 28.4
)
22.7
=c=
U iifc rm
T emperatu re
(4)
A ssumpti on
I (valid
kr:17.0
(3)
11.3
(2)
5.7
(1)
0
(200)
(180)
(160)
(140)
(120)
(100)
(80)
93
82
71
60
49
38
27
Sphere Temperature (°F) °C
MEASURED TEMPERATURE DEPENDENCE OF THE
SPHERE CONVECTION FILM COEFFICIENT
Figure 20
73
film coefficient as a function of sphere temperature.
The temperature
region in which the uniform body temperature assumption is invalid for
the sphere is shown in Figure 20.
Due to the error encountered at the near ambient temperatures,
the data for the flat plate were evaluated in the temperature region
of 49 to 38°C.
The lower limit of 38°C allows adequate time for re-
producible results and avoids the measurement errors associated with
lower plate temperatures.
74
APPENDIX C
DETERMINATION OF INSULATION LOSSES
The thermal loss through the insulation block must be known as
accurately as possible to assure maximum system accuracy.
A tech-
nique was developed to measure the conduction losses as a function
of plate temperature.
This involved warming the aluminum plate to
121°C (250°F) and covering the aluminum plate and the upper surface
of the insulation block with a second insulation block of identical
construction.
ent.
The temperature decay was monitored from 121°C to ambi-
The resulting conduction loss profile was examined to determine
the best possible temperature region for data collection.
The mathematical equations which describe the situation are the
same as those developed in Section 2.4.1, with the exception of the
elimination of convection losses.
The resulting equation is shown in
equation (A-1) where K represents the total conduction losses for both
insulation blocks.
me
to
(A-1)
T0-c1.1
The value of K used in the data reduction is therefore equal to onehalf of the total conduction loss measured.
The measured decay
times, temperatures, and resulting calculated conductance rates for
the double insulation block assembly are shown in Table 6.
The cal-
culated conductance for the single insulation block apparatus used in
the experimental arrangement is shown in Table 7, and is graphically
shown in Figure 21.
75
TABLE 6
MEASURED DECAY TIMES AND CALCULATED CONDUCTANCE LOSSES
Calculated conductance Btu/hr°F
(two insulation blocks)
To,
T
min
start
ambient
stop
5
213.7
70.4
203.1
0.475
8
203.1
70.3
190.0
0.398
6
190.0
70.4
181.9
0.364
10
181.9
70.5
170.6
0.330
10
170.6
70.7
161.4
0.298
14
161.4
70.7
150.6
0.278
16
150.6
70.7
140.7
0.255
21
140.7
70.7
130.2
0.255
20
130.2
70.8
121.9
0.230
35
121.9
70.6
110.9
0.213
50
110.9
70.2
99.3
0.207
55
99.3
70.0
90.4
0.203
105
90.4
69.9
80.3
0.200
To
t
76
_.
.
TABLE 7
TOTAL MEASURED CONDUCTION LOSS THROUGH
INSULATION BLOCK ASSEMBLY
Temperature °C (°F)
Energy loss w/°C (Btu/hr °F)
98
(208)
.412
(.238)
91
(196)
.344
(.199)
86
(186)
.315
(.182)
80
(176)
.285
(.165)
74
(165)
.257
(.149)
68
(155)
.239
(.138)
63
(145)
.219
(.127)
57
(135)
.208
(.120)
52
(126)
.199
(.115)
47
(116)
.183
(.106)
41
(105)
.178
(.103)
35
(
95)
.176
(.102)
0.13
(0.25)
0.11
(0.20)
-41(
)100-1
Data Collection
Temperature Bard
0.08
(0.15)
Apparatus
Conductance
W/°C
(BTU/hr', F)
0.05
(0.10)
0
(90)32
(110)43
(130)54
(150)66
(170)78
Temperature(°F) °C
(190)88
TOTAL INSULATION BLOCK ASSEMBLY THERMAL CONDUCTANCE
AS A FUNCTION OF PLATE TEMPERATURE
Figure 21
(210)99
78
APPENDIX D
TABULATED DATA AND RESULTS
79
TABLE 8
MEASURED DATA
u
Temperatures °F
T,
To
t
oo
ft/sec
6.33
7.00
9.43
11.16
11.93
13.34
17.39
17.64
22.42
23.60
24.95
24.75
24.32
7.89
10.75
11.74
23.95
23.58
min
8
6
6
5
5
4
4
4
4
4
3
3
3
8
5
6
3
23.95
7.89
4
4
4
4
8
12.11
5
13.50
18.02
5
21.51
16.87
21.61
23.77
5.58
10.54
11.93
13.83
18.02
21.40
23.67
5.58
5.96
11.93
13.66
17.26
23.86
21.70
4
4
4
6
5
5
5
4
4
3
6
6
5
5
4
3
4
.
123.4
121.8
70.8
71.3
123.1
72.1
119.3
119.1
124.0
119.0
118.8
122.3
125.2
121.5
122.0
123.9
124.3
120.3
121.5
119.6
124.8
119.3
119.6
119.6
121.5
119.5
123.0
125.0
120.9
118.6
122.8
121.4
120.7
124.3
119.3
120.5
119.3
123.9
121.5
120.0
123.6
123.5
121.5
120.6
71.7
72.1
70.9
71.8
71.7
71.5
72.2
72.2
71.3
73.1
71.7
70.7
71.3
71.4
72.5
71.9
71.6
71.6
72.0
70.8
71.1
70.4
71.7
71.7
71.8
71.7
71.8
71.8
71.2
72.2
72.1
71.1
72.7
71.5
71.7
71.3
72.1
71.5
T
angle of
attack
100.5
103.7
100.2
99.3
99.0
102.6
97.7
97.2
96.6
97.5
100.0
99.6
101.9
99.8
101.3
99.3
101.7
101.3
98.5
101.0
97.8
zero °
97.1
45°
99.0
99.4
99.8
97.7
94.9
100.8
98.5
98.1
98.9
96.8
95.8
99.0
100.9
99.5
96.7
97.4
97.6
99.4
94.4
H
H
II
H
tI
H
it
Hi
H
H
II
H
30°
H
H
H
H
H
H
H
H
H
II
H
60°
H
H
H
II
H
H
90°
II
H
H
H
H
H
80
TABLE 9
CALCULATED J-FACTORS
wind velocity
Reynolds #
j-factor
6.33
7.00
9.43
11.16
11.93
13.34
17.39
17.64
22.42
23.60
24.95
24.75
24.32
7.89
10.75
11.74
13.34
23.95
23.58
21.51
16.87
23.95
7.89
12.11
13.50
18.02
36200
40000
53900
63800
68200
76300
99400
100900
128200
135000
142700
141500
139000
45100
61500
67100
76300
137000
135000
123000
96500
137000
45000
69000
77000
103000
124000
135000
31900
60300
68200
79100
103000
122000
135000
31900
34100
68200
78100
98700
136000
124000
0.00421
0.00395
0.00398
0.00370
0.00355
0.00368
0.00330
0.00333
0.00302
0.00301
0.00294
0.00302
0.00299
0.00372
0.00340
0.00313
0.00311
0.00248
0.00242
0.00257
0.00277
0.00242
0.00405
0.00343
0.00342
0.00329
0.00283
0.00284
0.00637
0.00447
0.00396
0.00365
0.00335
0.00322
0.00305
0.00646
0.00634
0.00419
0.00393
0.00381
0.00319
0.00338
21.61
23.77
5.58
10.54
11.93
13.83
18.02
21.40
23.67
5.58
5.96
11.93
13.66
17.26
23.86
21.70
angle of attack
zero attack
30°
45°
60°
90°
81
APPENDIX E
TABLES, CHARTS, AND GRAPHS
82
TABLE 10
BIOT NUMBERS FOR VARIOUS MATERIALS
Assumptions
2
h = 19 w/m °C
Q = 0.00406 m
k w/m°k
Biot Number
pure copper
377
0.00020
pure aluminum
205
0.00037
(;
aluminum alloy
7075 T6 Alcladv=I
131
0.00058
Material
pure iron
63
0.0012
83
18.3
( 60)
.
/
/
rate
Flow
1416
(
000)
1180
15.2
(50)
(
500)
.
0
E12.2
"a(40)
.L.
(2000)
4J
a)
Exit
4)
-rn'-Velocity
9.1
(8(30)
4-2
707
500) g
47
1.7(
LA.1
472
6.1
(20)
I 000)
3.0
(10)
235
(500)
Design
Exit Are.
0
0
(1)0.09
(3)0.28
(2)0.19
Exit Area (ft2)
m2
FAN PERFORMANCE
Figure 22
84
TABLE 11
PROPERTIES OF THE ALUMINUM PLATE
Alloy
7075 Alclad
Temper
T6
Thickness
.406 cm (.160 in)
Surface condition
As rolled
Density
2794 kg/m
Specific heat
962 j/kg°k (.230 Btu/lb°F) @ 100°C
3
(174 lb/ft
3
)
870 j/kg°k (.208 Btu/lb°F) @ 0°C
Thermal conductivity
131 w/m°k (76 Btu/hr ft°F) @ 25°C
Taken from Strength of Metal Aircraft Elements,
MIL-HDBK-5.
85
TABLE 12
PROPERTIES OF INSULATION BLOCK MATERIAL
(manufacturer specification)
Manufacturer
Polymer Building Systems, Inc.
6918 Gage Street
Riverside, California 92504
Product name
PBS 800-L75
Insulation material
closed cell rigid urethane
Thermal conductivity
as specified
.12 -
Density
2 lb/ft
Service temperature
-73°C to +121°C (-100°F to 250°F)
R value
3.57 per half-inch thick sheet
.14 Btu/hr ft2°F/in
3
0.09
(0.3)
Measurement
Uncertainty
m/s
0.06
(0.2)
(ft/sec)
0.03
(0.1)
0
0
1.5(5)
3.0(10)
4.5(15)
6.1(20)
Measured air stream velocity(ft/sec) m/s
WIND VELOCITY MEASUREMENT UNCERTAINTY
Figure 23
7.6(25)
87
TABLE 13
WIND VELOCITY DATA
Oregon State University Reactor Building
velocity measured in miles per hour
November 29, 1978 -time
speed
8:10
1
15
20
25
30
35
40
45
50
55
1
9:00
05
10
15
20
25
30
35
time
11:00
05
10
15
20
25
30
35
2
2
1
1
2
1
0
0
0
0
.5
speed
time
speed
3.5
3.5
2:00
10
10
10
10
10
12
3
3
3
3
2
2
40
3
45
50
55
3
12:00
4
4
4
3
4
10:00
2
05
10
15
20
25
30
35
40
45
50
55
05
10
15
2
3
1:00
7
20
25
30
2
8
8
8
10
35
2
40
45
50
2
2
05
10
15
20
25
30
35
3
40
55
3
45
50
55
1
1
2
2
1.5
40
1
45
50
55
3
3
3
3
1
1
4.5
5
5
5
5
5.5
6
6.2
6.5
11
11
10
12
10
11
10
05
10
15
20
25
30
35
40
45
50
55
12
13
13
13
13.5
12
3:00
11
05
10
15
12
20
25
30
35
40
45
50
55
11
11
12
11
11
12
12
12
10
10
4:00
9
05
10
15
20
11
9
9
8
88
Table 13, continued
-- November 30, 1978 -time
speed
time
8:05
2.5
2.6
2.6
2.5
3.6
11:00
10
15
20
25
30
35
40
45
50
55
5.1
7.0
8.0
7.5
12
12.5
20
25
30
35
40
45
50
10
8.2
8.2
11.5
11.2
12.5
11.0
11.2
10
10
55
11
10:00
10
9:00
05
15
05
10
15
20
25
30
35
40
45
50
55
speed
16
05
10
15
20
25
30
35
40
45
50
55
15.5
16
12:00
20
19
20
16.2
05
10
15
20
25
30
35
40
45
50
55
15
16
16.2
16.2
19
19
20
20
20
21
22
20
20
20
20
23
20
time
seed
2:00
7
05
10
15
20
25
30
35
40
45
50
55
3:00
6.5
9
12
12
11
10
10.5
9
6.5
7.5
10
9
05
9
10
16
16
15
12
15
18
15
20
25
30
35
40
45
50
21
15
17
12.5
14
12.5
15
13
15
15
12.5
12.5
15
16
1:00
05
10
15
20
25
30
35
40
45
50
55
19.5
20
20
20
19
19
16.2
20
17
16
14
10
Statistical Analysis
V = 14.23 ft/sec
a = 8.90 ft/sec